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A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.

%I A051635 #69 Jul 02 2025 16:01:58
%S A051635 3,7,13,19,23,31,43,47,61,73,83,89,103,109,113,131,139,151,167,181,
%T A051635 193,199,229,233,241,271,283,293,313,317,337,349,353,359,383,389,401,
%U A051635 409,421,433,443,449,463,467,491,503,509,523,547,571,577,601,619,643,647
%N A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.
%C A051635 Primes prime(n) such that prime(n)-prime(n-1) < prime(n+1)-prime(n). - _Juri-Stepan Gerasimov_, Jan 01 2011
%C A051635 a(n) < A051634(n). a(n) ~ 2*prime(n). - _Thomas Ordowski_, Jul 25 2012
%C A051635 The inequality above is false. The least counterexample is a(19799) = 496291 > A051634(19799) = 496283. - _Amiram Eldar_, Nov 26 2023
%C A051635 Erdős called a weak prime an "early prime." He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof (Kuperberg 1992). See A229832 for a stronger conjecture. - _Jonathan Sondow_, Oct 13 2013
%D A051635 A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000
%H A051635 Amiram Eldar, <a href="/A051635/b051635.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A051635 Greg Kuperberg, <a href="http://www.math.niu.edu/~rusin/known-math/93_back/prizes.erd">The Erdos kitty: At least $9050 in prizes!</a>, Newsgroups: rec.puzzles, sci.math, 1992. [Broken link]
%H A051635 Greg Kuperberg, <a href="/A051635/a051635.erd.txt">The Erdos kitty: At least $9050 in prizes!</a>, Newsgroups: rec.puzzles, sci.math, 1992. [Cached copy]
%H A051635 Wikipedia, <a href="https://en.wikipedia.org/wiki/Weak_prime">Weak prime</a>.
%F A051635 a(1) = A229832(1). - _Jonathan Sondow_, Oct 13 2013
%F A051635 Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - _Alain Rocchelli_, Mar 17 2024
%e A051635 7 belongs to the sequence because 7 < (5+11)/2.
%t A051635 Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]<(#[[1]]+#[[3]])/2 &]][[2]] (* _Vladimir Joseph Stephan Orlovsky_, May 01 2008 *)
%t A051635 p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p, 2]], 1]]]]
%o A051635 (PARI) p=2;q=3;forprime(r=5,1e3,if(2*q<p+r,print1(q", "));p=q;q=r) \\ _Charles R Greathouse IV_, Jul 25 2011
%o A051635 (Haskell)
%o A051635 a051635 n = a051635_list !! (n-1)
%o A051635 a051635_list = g a000040_list where
%o A051635    g (p:qs@(q:r:ps)) = if 2 * q < (p + r) then q : g qs else g qs
%o A051635 -- _Reinhard Zumkeller_, May 09 2013
%Y A051635 Cf. A006562, A051634, A229832.
%Y A051635 Subsequence of A178943.
%Y A051635 Cf. A225495 (multiplicative closure).
%K A051635 nice,nonn,easy
%O A051635 1,1
%A A051635 _Felice Russo_, Nov 15 1999
%E A051635 More terms from _James Sellers_